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An Introduction to Financial Option Valuation

Book Description

This is a lively textbook providing a solid introduction to financial option valuation for undergraduate students armed with a working knowledge of a first year calculus. Written in a series of short chapters, its self-contained treatment gives equal weight to applied mathematics, stochastics and computational algorithms. No prior background in probability, statistics or numerical analysis is required. Detailed derivations of both the basic asset price model and the Black-Scholes equation are provided along with a presentation of appropriate computational techniques including binomial, finite differences and in particular, variance reduction techniques for the Monte Carlo method. Each chapter comes complete with accompanying stand-alone MATLAB code listing to illustrate a key idea. Furthermore, the author has made heavy use of figures and examples, and has included computations based on real stock market data.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. List of illustrations
  8. Preface
  9. 1. Options
    1. 1.1 What are options?
    2. 1.2 Why do we study options?
    3. 1.3 How are options traded?
    4. 1.4 Typical option prices
    5. 1.5 Other financial derivatives
    6. 1.6 Notes and references
    7. 1.7 Program of Chapter 1 and walkthrough
  10. 2. Option valuation preliminaries
    1. 2.1 Motivation
    2. 2.2 Interest rates
    3. 2.3 Short selling
    4. 2.4 Arbitrage
    5. 2.5 Put–call parity
    6. 2.6 Upper and lower bounds on option values
    7. 2.7 Notes and references
    8. 2.8 Program of Chapter 2 and walkthrough
  11. 3. Random variables
    1. 3.1 Motivation
    2. 3.2 Random variables, probability and mean
    3. 3.3 Independence
    4. 3.4 Variance
    5. 3.5 Normal distribution
    6. 3.6 Central Limit Theorem
    7. 3.7 Notes and references
    8. 3.8 Program of Chapter 3 and walkthrough
  12. 4. Computer simulation
    1. 4.1 Motivation
    2. 4.2 Pseudo-random numbers
    3. 4.3 Statistical tests
    4. 4.4 Notes and references
    5. 4.5 Program of Chapter 4 and walkthrough
  13. 5. Asset price movement
    1. 5.1 Motivation
    2. 5.2 Efficient market hypothesis
    3. 5.3 Asset price data
    4. 5.4 Assumptions
    5. 5.5 Notes and references
    6. 5.6 Program of Chapter 5 and walkthrough
  14. 6. Asset price model: Part I
    1. 6.1 Motivation
    2. 6.2 Discrete asset model
    3. 6.3 Continuous asset model
    4. 6.4 Lognormal distribution
    5. 6.5 Features of the asset model
    6. 6.6 Notes and references
    7. 6.7 Program of Chapter 6 and walkthrough
  15. 7. Asset price model: Part II
    1. 7.1 Computing asset paths
    2. 7.2 Timescale invariance
    3. 7.3 Sum-of-square returns
    4. 7.4 Notes and references
    5. 7.5 Program of Chapter 7 and walkthrough
  16. 8. Black–Scholes PDE and formulas
    1. 8.1 Motivation
    2. 8.2 Sum-of-square increments for asset price
    3. 8.3 Hedging
    4. 8.4 Black–Scholes PDE
    5. 8.5 Black–Scholes formulas
    6. 8.6 Notes and references
    7. 8.7 Program of Chapter 8 and walkthrough
  17. 9. More on hedging
    1. 9.1 Motivation
    2. 9.2 Discrete hedging
    3. 9.3 Delta at expiry
    4. 9.4 Large-scale test
    5. 9.5 Long-Term Capital Management
    6. 9.6 Notes
    7. 9.7 Program of Chapter 9 and walkthrough
  18. 10. The Greeks
    1. 10.1 Motivation
    2. 10.2 The Greeks
    3. 10.3 Interpreting the Greeks
    4. 10.4 Black–Scholes PDE solution
    5. 10.5 Notes and references
    6. 10.6 Program of Chapter 10 and walkthrough
  19. 11. More on the Black–Scholes formulas
    1. 11.1 Motivation
    2. 11.2 Where is μ?
    3. 11.3 Time dependency
    4. 11.4 The big picture
    5. 11.5 Change of variables
    6. 11.6 Notes and references
    7. 11.7 Program of Chapter 11 and walkthrough
  20. 12. Risk neutrality
    1. 12.1 Motivation
    2. 12.2 Expected payoff
    3. 12.3 Risk neutrality
    4. 12.4 Notes and references
    5. 12.5 Program of Chapter 12 and walkthrough
  21. 13. Solving a nonlinear equation
    1. 13.1 Motivation
    2. 13.2 General problem
    3. 13.3 Bisection
    4. 13.4 Newton
    5. 13.5 Further practical issues
    6. 13.6 Notes and references
    7. 13.7 Program of Chapter 13 and walkthrough
  22. 14. Implied volatility
    1. 14.1 Motivation
    2. 14.2 Implied volatility
    3. 14.3 Option value as a function of volatility
    4. 14.4 Bisection and Newton
    5. 14.5 Implied volatility with real data
    6. 14.6 Notes and references
    7. 14.7 Program of Chapter 14 and walkthrough
  23. 15. Monte Carlo method
    1. 15.1 Motivation
    2. 15.2 Monte Carlo
    3. 15.3 Monte Carlo for option valuation
    4. 15.4 Monte Carlo for Greeks
    5. 15.5 Notes and references
    6. 15.6 Program of Chapter 15 and walkthrough
  24. 16. Binomial method
    1. 16.1 Motivation
    2. 16.2 Method
    3. 16.3 Deriving the parameters
    4. 16.4 Binomial method in practice
    5. 16.5 Notes and references
    6. 16.6 Program of Chapter 16 and walkthrough
  25. 17. Cash-or-nothing options
    1. 17.1 Motivation
    2. 17.2 Cash-or-nothing options
    3. 17.3 Black–Scholes for cash-or-nothing options
    4. 17.4 Delta behaviour
    5. 17.5 Risk neutrality for cash-or-nothing options
    6. 17.6 Notes and references
    7. 17.7 Program of Chapter 17 and walkthrough
  26. 18. American options
    1. 18.1 Motivation
    2. 18.2 American call and put
    3. 18.3 Black–Scholes for American options
    4. 18.4 Binomial method for an American put
    5. 18.5 Optimal exercise boundary
    6. 18.6 Monte Carlo for an American put
    7. 18.7 Notes and references
    8. 18.8 Program of Chapter 18 and walkthrough
  27. 19. Exotic options
    1. 19.1 Motivation
    2. 19.2 Barrier options
    3. 19.3 Lookback options
    4. 19.4 Asian options
    5. 19.5 Bermudan and shout options
    6. 19.6 Monte Carlo and binomial for exotics
    7. 19.7 Notes and references
    8. 19.8 Program of Chapter 19 and walkthrough
  28. 20. Historical volatility
    1. 20.1 Motivation
    2. 20.2 Monte Carlo type estimates
    3. 20.4 Maximum likelihood estimate
    4. 20.5 Other volatility estimates
    5. 20.6 Example with real data
    6. 20.7 Notes and references
    7. 20.8 Program of Chapter 20 and walkthrough
  29. 21. Monte Carlo Part II: variance reduction by antithetic variates
    1. 21.1 Motivation
    2. 21.2 The big picture
    3. 21.3 Dependence
    4. 21.4 Antithetic variates: uniform example
    5. 21.5 Analysis of the uniform case
    6. 21.6 Normal case
    7. 21.7 Multivariate case
    8. 21.8 Antithetic variates in option valuation
    9. 21.9 Notes and references
    10. 21.10 Program of Chapter 21 and walkthrough
  30. 22. Monte Carlo Part III: variance reduction by control variates
    1. 22.1 Motivation
    2. 22.2 Control variates
    3. 22.3 Control variates in option valuation
    4. 22.4 Notes and references
    5. 22.5 Program of Chapter 22 and walkthrough
  31. 23. Finite difference methods
    1. 23.1 Motivation
    2. 23.2 Finite difference operators
    3. 23.3 Heat equation
    4. 23.4 Discretization
    5. 23.5 FTCS and BTCS
    6. 23.6 Local accuracy
    7. 23.7 Von Neumann stability and convergence
    8. 23.8 Crank–Nicolson
    9. 23.9 Notes and references
    10. 23.10 Program of Chapter 23 and walkthrough
  32. 24. Finite difference methods for the Black–Scholes PDE
    1. 24.1 Motivation
    2. 24.2 FTCS, BTCS and Crank–Nicolson for Black–Scholes
    3. 24.3 Down-and-out call example
    4. 24.4 Binomial method as finite differences
    5. 24.5 Notes and references
    6. 24.6 Program of Chapter 24 and walkthrough
  33. References
  34. Index